Graph Modification Problems Research Articles

Graph Modification for Edge-Coloured and Signed Graph Homomorphism Problems: Parameterized and Classical Complexity

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from edge-coloured graph $G$ to edge-coloured graph $H$ is a vertex-mapping from $G$ to $H$ that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph $H$. The question we are interested in is: given an edge-coloured graph $G$, can we perform $k$ graph operations so that the resulting graph admits a homomorphism to $H$? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs $+$ and $-$. We denote the corresponding problems (parameterized by $k$) by VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. These problems generalise $H$-COLOURING (to decide if an input graph admits a homomorphism to a fixed target $H$). Our main focus is when $H$ is an edge-coloured graph with at most two vertices, a case that is already interesting as it includes problems such as VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph $H$, we give a P/NP-c complexity dichotomy for VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. We then address their parameterized complexity. We show that VD-$H$-COLOURING and ED-$H$-COLOURING for all such $H$ are FPT. In contrast, already for some $H$ of order 3, unless P=NP, none of the three problems is in XP, since 3-COLOURING is NP-c. We show that SW-$H$-COLOURING is different: there are three 2-edge-coloured graphs $H$ of order 2 for which SW-$H$-COLOURING is W-hard, and assuming the ETH, admits no algorithm in time $f(k)n^{o(k)}$. For the other cases, SW-$H$-COLOURING is FPT.

On the threshold of intractability

The computational complexity of the graph modification problems Threshold Editing and Chain Editing has been an important open question in computational graph theory for more than 15 years. These problems consist of adding and deleting the fewest possible edges to transform a graph into a threshold or chain graph. We show that both problems are NP-hard, resolving a conjecture by Natanzon, Shamir, and Sharan from 2001. On the positive side, we show both problems admit quadratic vertex kernels and give subexponential time parameterized algorithms solving both problems in 2O(klog⁡k)+nO(1) time. Few natural problems are known to be in this complexity class. These results also extend to the completion and deletion variants of both problems, and threshold/chain graphs are the only known graph classes for which all three versions—completion, deletion, and editing—are both NP-complete and solvable in subexponential parameterized time.

Parameterized Dynamic Cluster Editing

We introduce a dynamic version of the NP-hard graph modification problem Cluster Editing. The essential point here is to take into account dynamically evolving input graphs: having a cluster graph (that is, a disjoint union of cliques) constituting a solution for a first input graph, can we cost-efficiently transform it into a “similar” cluster graph that is a solution for a second (“subsequent”) input graph? This model is motivated by several application scenarios, including incremental clustering, the search for compromise clusterings, or also local search in graph-based data clustering. We thoroughly study six problem variants (three modification scenarios edge editing, edge deletion, edge insertion; each combined with two distance measures between cluster graphs). We obtain both fixed-parameter tractability as well as (parameterized) hardness results, thus (except for three open questions) providing a fairly complete picture of the parameterized computational complexity landscape under the two perhaps most natural parameterizations: the distances of the new “similar” cluster graph to (1) the second input graph and to (2) the input cluster graph.

Observability Properties of Colored Graphs

A colored graph is a directed graph in which nodes or edges have been assigned colors that are not necessarily unique. Observability problems in such graphs consider whether an agent observing the colors of edges or nodes traversed on a path in the graph can determine which node they are at currently or which nodes were visited earlier in the traversal. Previous research efforts have identified several different notions of observability as well as the associated properties of graphs for which those observability properties hold. This paper unifies the prior work into a common framework with several new results about relationships between those notions and associated graph properties. The new framework provides an intuitive way to reason about the attainable accuracy as a function of lag and time spent observing, and identifies simple modifications to improve the observability of a given graph. We show that one form of the graph modification problem is in NP-Complete. The intuition of the new framework is borne out with numerical experiments. This work has implications for problems that can be described in terms of an agent traversing a colored graph, including the reconstruction of hidden states in a hidden Markov model (HMM).

On the Parameterized Complexity of Graph Modification to First-Order Logic Properties

We establish connections between parameterized/kernelization complexity of graph modification problems and expressibility in logic. For a first-order logic formula φ, we consider the problem of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the resulting modification has the property expressible by φ. We provide sufficient and necessary conditions on the structure of the prefix of φ specifying when the corresponding graph modification problem is fixed-parameter tractable (parameterized by k) and when it admits a polynomial kernel.

Subexponential Parameterized Algorithm for I nterval C ompletion

In the I nterval C ompletion problem we are given an n -vertex graph G and an integer k , and the task is to transform G by making use of at most k edge additions into an interval graph. This is a fundamental graph modification problem with applications in sparse matrix multiplication and molecular biology. The question about fixed-parameter tractability of I nterval C ompletion was asked by Kaplan et al. [FOCS 1994; SIAM J. Comput. 1999] and was answered affirmatively more than a decade later by Villanger et al. [STOC 2007; SIAM J. Comput. 2009], who presented an algorithm with running time O ( k 2 k n 3 m ). We give the first subexponential parameterized algorithm solving I nterval C ompletion in time k O (√ k ) n O (1) . This adds I nterval C ompletion to a very small list of parameterized graph modification problems solvable in subexponential time.

Rank Reduction of Oriented Graphs by Vertex and Edge Deletions

In this paper we continue our study of graph modification problems defined by reducing the rank of the adjacency matrix of the given graph, and extend our results from undirected graphs to modifying the rank of skew-adjacency matrix of oriented graphs. An instance of a graph modification problem takes as input a graph G and a positive integer k, and the objective is to either delete k vertices/edges or edit k edges so that the resulting graph belongs to a particular family $$\mathcal{F}$$ of graphs. Given a fixed positive integer r, we define $$\mathcal{F}_r$$ as the family of oriented graphs where for each $$G\in \mathcal{F}_r$$ , the rank of the skew-adjacency matrix of G is at most r. Using the family $$\mathcal{F}_r$$ we do algorithmic study, both in classical and parameterized complexity, of the following graph modification problems: $$r$$ -Rank Vertex Deletion, $$r$$ -Rank Edge Deletion. We first show that both the problems are NP-Complete. Then we show that these problems are fixed parameter tractable (FPT) by designing an algorithm with running time $$2^{\mathcal{O}(k \log r)}n^{\mathcal{O}(1)}$$ for $$r$$ -Rank Vertex Deletion, and an algorithm for $$r$$ -Rank Edge Deletion running in time $$2^{\mathcal{O}(f(r) \sqrt{k} \log k )}n^{\mathcal{O}(1)}$$ . In addition to our FPT results we design polynomial kernels for these problems. Our main structural result, which is the fulcrum of all our algorithmic results, is that for a fixed integer r the size of any “reduced graph” in $$\mathcal{F}_r$$ is upper bounded by $$3^r$$ . This result is of independent interest and generalizes a similar result of Kotlov and Lovasz regarding reduced oriented graphs of rank r.

Editing to a planar graph of given degrees

We consider the following graph modification problem. Let the input consist of a graph G=(V,E), a weight function w:V∪E→N, a cost function c:V∪E→N0 and a degree function δ:V→N0, together with three integers kv, ke and C. The question is whether we can delete a set of vertices of total weight at most kv and a set of edges of total weight at most ke so that the total cost of the deleted elements is at most C and every non-deleted vertex v has degree δ(v) in the resulting graph G′. We also consider the variant in which G′ must be connected. Both problems are known to be NP-complete and W[1]-hard when parameterized by kv+ke. We prove that, when restricted to planar graphs, they stay NP-complete but have polynomial kernels when parameterized by kv+ke.

An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour (J Comb Theory Ser B 96(4):514–528, 2006). Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an n-vertex graph G and a positive integer k, we want to decide whether there is a set of at most k vertices whose removal turns G into a graph of linear rankwidth at most 1 and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time $$f(k)\cdot n^3$$ for some function f, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time $$8^k\cdot n^{{\mathcal {O}}(1)}$$ . The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time $$2^{{\mathcal {O}}(k)}\cdot n^4$$ . We also prove that the running time cannot be improved to $$2^{o(k)}\cdot n^{{\mathcal {O}}(1)}$$ under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.

Graph modification problem for some classes of graphs

The issue of determining the complexity of modification problems for chordal bipartite graphs has been raised multiple times in the literature. We show that the completion and deletion problems for chordal bipartite graphs are NP-hard. The corresponding problems for weakly chordal graphs are already known to be hard. As a byproduct, we obtain results on modification problems for a variety of related classes of graphs and also simplify the arguments for the class of weakly chordal graphs.

Win-win kernelization for degree sequence completion problems

We study provably effective and efficient data reduction for a class of NP-hard graph modification problems based on vertex degree properties. We show fixed-parameter tractability for NP-hard graph completion (that is, edge addition) cases while we show that there is no hope to achieve analogous results for the corresponding vertex or edge deletion versions. Our algorithms are based on transforming graph completion problems into efficiently solvable number problems and exploiting f-factor computations for translating the results back into the graph setting. Our core observation is that we encounter a win-win situation: either the number of edge additions is small or the problem is polynomial-time solvable. This approach helps in answering an open question by Mathieson and Szeider [JCSS 2012] concerning the polynomial kernelizability of Degree Constraint Edge Addition and leads to a general method of approaching polynomial-time preprocessing for a wider class of degree sequence completion problems.

Reducing rank of the adjacency matrix by graph modification

The main topic of this article is to study a class of graph modification problems. A typical graph modification problem takes as input a graph G, a positive integer k and the objective is to add/delete k vertices (edges) so that the resulting graph belongs to a particular family, F, of graphs. In general the family F is defined by forbidden subgraph/minor characterization. In this paper rather than taking a structural route to define F, we take algebraic route. More formally, given a fixed positive integer r, we define Fr as the family of graphs where for each G∈Fr, the rank of the adjacency matrix of G is at most r. Using the family Fr we initiate algorithmic study, both in classical and parameterized complexity, of following graph modification problems: r-Rank Vertex Deletion, r-Rank Edge Deletion and r-Rank Editing. These problems generalize the classical Vertex Cover problem and a variant of the d-Cluster Editing problem. We first show that all the three problems are NP-Complete. Then we show that these problems are fixed parameter tractable (FPT) by designing an algorithm with running time 2O(klog⁡r)nO(1) for r-Rank Vertex Deletion, and an algorithm for r-Rank Edge Deletion and r-Rank Editing running in time 2O(f(r)klog⁡k)nO(1). We complement our FPT result by designing polynomial kernels for these problems.

Chordal Editing is Fixed-Parameter Tractable

Graph modification problems are typically asked as follows: is there a small set of operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem which allows all three types of operations: given a graph $G$ and integers $k_1$, $k_2$, and $k_3$, the \textsc{chordal editing} problem asks whether $G$ can be transformed into a chordal graph by at most $k_1$ vertex deletions, $k_2$ edge deletions, and $k_3$ edge additions. Clearly, this problem generalizes both \textsc{chordal vertex/edge deletion} and \textsc{chordal completion} (also known as \textsc{minimum fill-in}). Our main result is an algorithm for \textsc{chordal editing} in time $2^{O(k\log k)}\cdot n^{O(1)}$, where $k:=k_1+k_2+k_3$ and $n$ is the number of vertices of $G$. Therefore, the problem is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm is both more efficient and conceptually simpler than the previously known algorithm for the special case \textsc{chordal deletion}.

An overview of kernelization algorithms for graph modification problems

Kernelization algorithms for graph modification problems are important ingredients in parameterized computation theory. In this paper, we survey the kernelization algorithms for four types of graph modification problems, which include vertex deletion problems, edge editing problems, edge deletion problems, and edge completion problems. For each type of problem, we outline typical examples together with recent results, analyze the main techniques, and provide some suggestions for future research in this field.

Editing Simple Graphs

We study the complexity of turning a given graph, by edge editing, into a target graph whose critical-clique graph is any fixed graph. The problem came up in practice, in an effort of mining huge word similarity graphs for well structured word clusters. It also adds to the rich field of graph modification problems. We show in a generic way that several variants of this problem are in SUBEPT. As a special case, we give a tight time bound for edge deletion to obtain a single clique and isolated vertices, and we round up this study with NP-completeness results for a number of target graphs.

Contracting chordal graphs and bipartite graphs to paths and trees

We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems, which we call Tree Contraction and Path Contraction, respectively, are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n+m) and O(nm) time, respectively. As a contrast, both problems remain NP-complete when restricted to bipartite input graphs.

BOUNDED SEARCH TREE ALGORITHMS FOR PARAMETRIZED COGRAPH DELETION: EFFICIENT BRANCHING RULES BY EXPLOITING STRUCTURES OF SPECIAL GRAPH CLASSES

Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of P4-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parametrized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [R. Niedermeier and P. Rossmanith, An efficient fixed-parameter algorithm for 3-hitting set, J. Discrete Algorithms1(1) (2003) 89–102] and computer-aided branching rules [J. Gramm, J. Guo, F. Hüffner and R. Niedermeier, Automated generation of search tree algorithms for hard graph modification problems, Algorithmica39(4) (2004) 321–347].

On the (Non-)Existence of Polynomial Kernels for P l -Free Edge Modification Problems

Given a graph G=(V,E) and a positive integer k, an edge modification problem for a graph property ź consists in deciding whether there exists a set F of pairs of V of size at most k such that the graph $H=(V,E\vartriangle F)$ satisfies the property ź. In the źedge-completion problem, the set F is constrained to be disjoint from E; in the źedge-deletion problem, F is a subset of E; no constraint is imposed on F in the źedge-editing problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity (Cai in Inf. Process. Lett. 58:171---176, 1996; Fellows et al. in FCT, pp. 312---321, 2007; Heggernes et al. in STOC, pp. 374---381, 2007). When parameterized by the size k of the set F, it has been proved that if ź is a hereditary property characterized by a finite set of forbidden induced subgraphs, then the three ź edge-modification problems are FPT (Cai in Inf. Process. Lett. 58:171---176, 1996). It was then natural to ask (Bodlaender et al. in IWPEC, 2006) whether these problems also admit a polynomial kernel. in polynomial time to an equivalent instance (Gź,kź) with size bounded by a polynomial in k). Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively (Kratsch and Wahlstrom in IWPEC, pp. 264---275, 2009). However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. question to characterize for which type of graph properties, the parameterized edge-modification problems have polynomial kernels. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that Parameterized cograph edge-modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free edge-deletion and the Cl-free edge-deletion problems for lź7 and lź4 respectively. Indeed, if they exist, then NP⊆coNP/poly.

Graph-based data clustering with overlaps

We introduce overlap cluster graph modification problems where, other than in most previous works, the clusters of the target graph may overlap. More precisely, the studied graph problems ask for a minimum number of edge modifications such that the resulting graph consists of clusters (that is, maximal cliques) that may overlap up to a certain amount specified by the overlap number s . In the case of s -vertex-overlap, each vertex may be part of at most s maximal cliques; s -edge-overlap is analogously defined in terms of edges. We provide a complexity dichotomy (polynomial-time solvable versus NP-hard) for the underlying edge modification problems, develop forbidden subgraph characterizations of “cluster graphs with overlaps”, and study the parameterized complexity in terms of the number of allowed edge modifications, achieving fixed-parameter tractability (in case of constant s -values) and parameterized hardness (in case of unbounded s -values).

Polynomial kernels for 3-leaf power graph modification problems

A graph G = ( V , E ) is a 3-leaf power iff there exists a tree T the leaf set of which is V and such that u v ∈ E iff u and v are at distance at most 3 in T . The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3- leaf power), completion and edge-deletion are FPT when parameterized by the size of the edge set modification. However, polynomial kernels were known for none of these three problems. For each of them, we provide kernels with O ( k 3 ) vertices that can be computed in linear time. We thereby answer an open problem first mentioned by Dom et al. (2004) [8].